CP Algebra 2. Unit 3B: Polynomials. Name: Period:

Transcription

1 CP Algebra 2 Unit 3B: Polynomials Name: Period:

2 Learning Targets 10. I can use the fundamental theorem of algebra to find the expected number of roots. Solving Polynomials 11. I can solve polynomials by graphing (with a calculator). 12. I can solve polynomials by factoring. Finding and Using Roots 13. I can find all of the roots of a polynomial. 14. I can write a polynomial function from its complex roots. Graphing 15. I can graph polynomials.

3 Solving Polynomials After this lesson and practice, I will be able to! use the fundamental theorem of algebra to find the expected number of roots. (LT 10)! solve polynomials by graphing (with a calculator). (LT 11)! solve polynomials by factoring. (LT 12) In the quadratics unit, you learned five strategies for solving quadratic equations. Let s see how many you can remember! 1) 4) 2) 5) 3) Today we re going to solve polynomials, which will seem very similar to solving quadratics. There s one thing we should learn first that will help us as we solve Find the Expected Number of Roots (LT 10) Look back at the chart you filled out at the beginning of this unit. How does the degree of the polynomial relate to the number of x-intercepts? The number of to a polynomial function is equal to the of the polynomial. This observation is a very important fact in algebra (Corollary to) The Fundamental Theorem of Algebra Every polynomial in one variable of degree n > 0 exactly zeros, including and zeros. has This theorem makes it possible to know the number and type of zeros in a given function, which can be helpful in finding all zeros of a polynomial. Example 1: Determine the number of zeros of the polynomial. a.! f (x)= x 3 2x 2 + 4x 8 b. y = 15x!" + 3x! 9 You can always use that trick to figure out how many zeros you should expect from a polynomial. Now let s solve!

4 Today we ll be solving by factoring and graphing. Let s start with graphs, since it s basically the same process as when we solved quadratics by graphing. Solving by Graphing (LT 11) Our graphing calculators will help us find zeros of a polynomial function. Let s use y = x! + 12x! + x 1 1) Enter the equation in your calculator as Y 1 =. Press GRAPH. 2) To make sure we can see the graph, click ZOOM and ZStandard or ZoomFit You should see a skinny parabola that looks like it has two zeros. But let s use our Fundamental Theorem of Algebra trick to make sure there are only two zeros Based on the Fundamental Theorem of Algebra, how many zeros should this polynomial have? Let s edit the window until we can see all zeros. Then continue on with step 3. 3) Press 2 nd TRACE, then press 2: ZERO. 4) Move your cursor just to the left of the first point of intersection. Press ENTER. 5) Move your cursor just to the right of the first point of intersection. Press ENTER. 6) The screen will show Guess. Press ENTER again. The calculator will display the zero. 7) Repeat steps 3-6 to obtain the rest of the zeros. Example 2: Find the expected number of zeros, then use your graphing calculator to find the zeros of the function y = x! + 12x! + x 1 (Hint: You may need to zoom out!) Example 3: Find the expected number of zeros, then use your graphing calculator to find the zeros of the function y = 2x! + x 7. Example 4: Find the expected number of zeros, then use your graphing calculator to find the zeros of the function y = x! + 2x! 6x! 2.

5 There s another method of solving that should be pretty familiar to you by now! Solving By Factoring (LT 12) Recall our previous strategies for factoring quadratics: 1) 4) 2) 5) 3) Let s add two additional factoring strategies before learning how to use them to solve polynomial equations. Polynomial Factoring Strategy #1: Sums and Differences of Cubes Just as there are patterns for the difference of two, there also exist patterns for the sums and differences of two cubes! Sum of Two Cubes Difference of Two Cubes Example 5: Factor completely. 3 a. x + 64 b. 16z 250z 5 2 Example 6: Factor completely. 3 a. x + 8 b. 3 8x 1 c m 216

6 Polynomial Factoring Strategy #2: Quadratic Forms Some polynomials of higher-degree can be solved using strategies you used when you factored quadratics. The key is in recognizing if the polynomial is in quadratic form. Example 7: Factor completely. a.!x 4 2x 2 8 b.!x 4 +7x 2 +6 Example 8: Factor completely. a.!x 4 x 2 2 b.!x 4 +8x 2 9 Now that you have some additional factoring strategies, let s utilize these strategies to solve polynomial equations by factoring! Example 9: Find the expected number of zeros, then solve each equation by factoring. a.!27x 3 +1 = 0 b.!x 4 x 2 = 12 c.!3x 3 +2x 2 15x 10 = 0 Expected #: Expected #: Expected #:

7 Finding and Using Roots After this lesson and practice, I will be able to! find all of the roots of a polynomial. (LT 13)! write a polynomial function from its complex roots. (LT 14) Today we re going to learn a few other techniques for finding roots and using them to write equations in factored form. Identify Roots (LT 13) You can identify any rational roots by graphing a polynomial in your calculator and using the zero function to find a root. Once you have a root, you can use synthetic division to get the polynomial down to a quadratic. See the box below for the steps. Example 3: Find all roots of each function and write each function in factored form with integer coefficients. a.! f (x)= x 3 7x 2 +2x + 40 Strategies for Finding All Roots of a Polynomial 1) List all possible rational roots. 2) Use your calculator to verify one rational root. 3) Use synthetic division until the expression is quadratic and then use other algebraic techniques to find the remaining zeros. b.! f (x)= 2x 3 5x 2 14x +8 Example 4: Find all roots of the function! f (x)= 2x 3 +3x 2 8x +3 and write it in factored form with integer coefficients.

8 Unfortunately, as you have observed, not all polynomials have exclusively roots. Nevertheless, you can use rational roots to help you find all zeros of a polynomial. Example 5: Find all roots of each function and write each function in factored form. a.! f (x)= x 4 5x 3 11x 2 +25x +30 b.! f (x)= 3x 3 + x 2 x +1 The results to these examples lead us to two additional polynomial theorems: Irrational Root Theorem If is a root of a polynomial equation with rational coefficients, then the is also a root of the equation. Imaginary Root Theorem If is a root of a polynomial equation with real coefficients, then the is also a root of the equation. NAME THAT CONJUGATE! 1.!3 7 2.!1+2i 3.! 12 5i 4.! 15 5.!πi Example 6: Suppose a polynomial with rational coefficients has the following roots:! and! 4 2. Find two additional roots. Example 7: A quartic polynomial with real coefficients has roots of -3 and!2 5i. Which of the following cannot be another root of the polynomial? A. 12 B. 0 C.! 2 D.!2+5i

9 Example 8: Find all roots of the function! f (x)= x 3 2x 2 3x +10 and write it in factored form. Write Polynomials From Complex Roots (LT 14) Now we ll explore how to write polynomial equations using information about its roots. Example 9: Find a polynomial function in standard form whose graph has x-intercepts 3, 5, -4, and y-intercept 180. Recall from the previous lesson, that when polynomials have or zeros, they always appear as. Example 10: Write a polynomial function in standard form with real coefficients and zeros x= 2, x= 5, x= 3+ 4i.

10 Graphing Polynomials After this lesson and practice, I will be able to! graph polynomials. (LT 15) Let s combine everything we ve learned to graph some polynomials! Example 1: Find all zeros of! f (x)= x 3 +3x 2 x 3. Then complete the requested information: Zeros/Roots: Factored Form:! f (x)= f( ) =, f( ) =, f( ) = because y-intercept = (, ) x-intercept(s) = (, ) (, ) (, ) End behavior: as as